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Metamath Proof Explorer

Theorem ovigg

Description: The value of an operation class abstraction. Compared with ovig , the condition ( x e. R /\ y e. S ) is removed. (Contributed by FL, 24-Mar-2007) (Revised by Mario Carneiro, 19-Dec-2013)

		Ref	Expression
	Hypotheses	ovigg.1	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ) → ( 𝜑 ↔ 𝜓 ) )
		ovigg.4	⊢ ∃* 𝑧 𝜑
		ovigg.5	⊢ 𝐹 = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 }
	Assertion	ovigg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → ( 𝜓 → ( 𝐴 𝐹 𝐵 ) = 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		ovigg.1	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ) → ( 𝜑 ↔ 𝜓 ) )
2		ovigg.4	⊢ ∃* 𝑧 𝜑
3		ovigg.5	⊢ 𝐹 = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 }
4	1	eloprabga	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → ( ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ ∈ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } ↔ 𝜓 ) )
5		df-ov	⊢ ( 𝐴 𝐹 𝐵 ) = ( 𝐹 ‘ ⟨ 𝐴 , 𝐵 ⟩ )
6	3	fveq1i	⊢ ( 𝐹 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } ‘ ⟨ 𝐴 , 𝐵 ⟩ )
7	5 6	eqtri	⊢ ( 𝐴 𝐹 𝐵 ) = ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } ‘ ⟨ 𝐴 , 𝐵 ⟩ )
8	2	funoprab	⊢ Fun { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 }
9		funopfv	⊢ ( Fun { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } → ( ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ ∈ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } → ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = 𝐶 ) )
10	8 9	ax-mp	⊢ ( ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ ∈ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } → ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = 𝐶 )
11	7 10	eqtrid	⊢ ( ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ ∈ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } → ( 𝐴 𝐹 𝐵 ) = 𝐶 )
12	4 11	biimtrrdi	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → ( 𝜓 → ( 𝐴 𝐹 𝐵 ) = 𝐶 ) )